A quadrat is a fixed and easily repeatable constant area (normally 1 metre squared), used where there is no environmental gradient.
A transect uses a quadrat in a line along an environmental gradient.
Random sampling is used with quadrats to remove observer bias and as you are only finding the abundance of a species in a particular area.
Systematic sampling is used along transects as you are also wanting to find the distribution of the species as well as the abundance.